# UK Lotto Odds -- Lottery Probability

TR Smith is a product designer and former teacher who uses math in her work every day.

In the United Kingdom the biggest lottery after EuroMillions is the "Lotto," an easy-to-play random drawing game with jackpots in the several millions of pounds. A play slip for UK Lotto costs £2 and comes with automatic entry into Lotto Raffle, a separate drawing that awards £20,000 to at least 50 lucky players every Wednesday and Saturday during the official Lotto draw. Players can choose their own numbers, or let the lottery computer choose for them, known as "lucky dip."

If the lottery jackpot is not won during a particular drawing, the prize amount rolls over to the next drawing, so jackpots can grow quite large if there are many non-winners in a row. Rollovers also increase the number of Lotto Raffle drawings.

## How to Play UK Lotto

Lotto has a 6-out-of-49 structure like many other lottery games, such as Canadian 6/49, Ohio Classic Lotto, and New Jersey Pick-6. On your ticket you simply select 6 distinct numbers from between 1 and 49. On the day of the drawing, lottery officials will select 6 numbered balls from the machine, and a 7th bonus ball that is only used for the second prize level. You do not select a 7th bonus number on your ticket.

## Probability of Winning UK Lotto

6 Out of 6: There is only one way to match all 6 out of 6 numbers in the drawing, and the total number of lottery ticket combinations is (49 C 6) = 13,983,816, where (x C y) is the combination function. Therefore, the probability of winning a share of the jackpot is the reciprocal of this number,

1/13983816
= 0.000000071511
= 0.0000071511%

5 Out of 5 + Bonus: The next prize level is for matching 5 out of 6 numbers plus the bonus number. The number of ways this can occur is (6 C 5)*1 = 6, therefore the probability is 6 divided by the total number of combinations,

6/13983816
= 1/2330636
= 0.00000042907
= 0.000042907%

5 Out of 5: The third prize level is matching 5 out of 6 but not the bonus number. The number of ways this can occur is (6 C 5)*42 = 252, therefore the probability is

252/13983816
= 0.000018021
= 0.0018021%
≈ odds of 1 in 55491

4 Out of 6: The fourth prize level of UK Lotto is awarded to players whose play slips match 4 out of 6 numbers. The number of ways to achieve this is (6 C 4)*(43 C 2). This is the number of ways to choose 4 numbers correctly out of 6, times the number of ways to choose 2 incorrect numbers out of 43. This is

15*903/13983816
= 13545/13983816
= 0.00096861972
= 0.096861972%
≈ odds of 1 in 1032

3 Out of 6: The lowest prize is given if you match 3 out of 6 numbers. The number of ways this can happen is (6 C 3)*(43 C 3). The probability is

20*12341/13983816
= 246820/13983816
= 0.01765040386
= 1.765040386%
≈ odds of 1 out of 57

Here is a table summary of the odds and payouts for all prize levels of the UK Lotto game.

Type of Match
Cash Prize
Odds
6/6
52% of payout fund
1 in 13,983,816
5/6 + bonus
16% of payout fund
1 in 2,330,636
5/6
10% of payout fund
1 in 55,491
4/6
22% of payout fund
1 in 1,032
3/6
£25
1 in 57

## Lotto Hot Picks

Lotto Hot Picks is an associated game that use the main Lotto drawing for its winning numbers. A Lotto Hot Picks wager is £1 per board, i.e, per set of numbers selected. To play, you first decide if you want to bet on the outcome of 1, 2, 3, 4, or 5 of the winning numbers for the upcoming Lotto drawing. If you guess all of the numbers correctly, you win a cash prize proportional to the odds. It is similar to keno.

For example, if you play 1 number (1-spot game) and guess it correctly, the odds of this happening are about 1 in 8 and you win £5. If you play a set of 5 numbers (5-spot game) and guess them all correctly, the odds of this happening are 1 in 317,814 and you win £130,000. There are no prizes for partially matching the numbers. Your numbers must match a subset of the 6 main numbers of the Lotto drawing, not the bonus number.

Here are the odds and payouts for the UK Lotto Hot Picks game.

Type of Wager
Odds of Guessing Corrrectly
Prize
guess 5 numbers
1 in 317,814
£130,000
guess 4 numbers
1 in 14,125
£7,000
guess 3 numbers
1 in 921
£450
guess 2 numbers
1 in 78
£40
guess 1 number
1 in 8
£5

In Lotto Hot Picks, the type of wager with the best expected return is the 1-spot game, with an expected return of £0.6122 per £1 wagered. The 5-spot game has the worst expected return, £0.409 per £1 wagered. The expected return is computed by multiplying the prize amount by the probability of winning.

The probability of winning the n-spot game is given by the fraction

(49-n C 6-n)/13983816

0

2

6

5

38

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